Reductions and necessary conditions for tall Borel Ramsey ideals

Abstract

An ideal $\mathcal{I}$ on $\omega$ has the Ramsey property if ${\mathcal{I}}^{+}\to ({\mathcal{I}}^{+}{)}_2^2$: every $2$-colouring of the pairs of an $\mathcal{I}$-positive set has an $\mathcal{I}$-positive homogeneous subset. Whether a tall Borel ideal can have the Ramsey property is an open question of Hrušák, Meza-Alcántara, Thümmel and Uzcátegui; a coanalytic example exists in ZFC, so a negative answer must use definability essentially. Our main theorem, a synthesis of the results of the paper, states that a tall Borel Ramsey ideal admits no countable local reading. Below every positive set, such an ideal is not a countable intersection of topologically represented (or tall analytic $P$-) ideals, and its quotient has no countable dense subset. Moreover, every quotient name for a new real has uncountable width, the colouring witnessing non-selectivity of the generic ultrafilter is never read continuously on a positive condition, and hereditary tall subfamilies saturate every finite window of barrier dimensions coherently but never all dimensions at once. We prove separately that a weakly selective $q^{+}$ ideal admits no positive ${\mathcal{E}\mathcal{D}}_{\mathrm{fin}}$-carrier. The converse question -- must Borelness force a properness-like countable reading on some positive condition? -- is stated in three precise forms with proved consequences: two of them would refute tall Borel Ramsey ideals outright, the third the strictly weaker Nash--Williams class. The main question remains open.

30 pages